Comment on 'On computing quantum waves exactly from classical action'

The comment is largely internally coherent. Its core thesis is stable throughout: substituting ψ_j = √ρ_j e^{iφ_j/ℏ} into the Schrödinger equation necessarily produces the quantum-potential term Q = -(ℏ²/2M)(∇²√ρ_j)/√ρ_j, so any derivation yielding only the classical Hamilton–Jacobi equation must have omitted spatial derivatives of the amplitude. Section 1 states this claim, Section 2 carries out the derivative expansion, and Section 4 returns to the same conclusion. The logical dependence is clear and consistent.

There are, however, a few overstatements and local ambiguities that keep the score below 5. First, the text says that the target derivation 'implicitly assumes ∇√ρ_j = 0' and equates this with a perfectly homogeneous density across all space. Mathematically, the condition needed to remove Q is weaker: ∇²√ρ_j = 0 (away from nodes/singularities), and there are nonconstant harmonic amplitudes such as √ρ ∝ 1/r in 3D free space. The author partly acknowledges this later in Section 3.1, which softens the issue, but this does create a local mismatch between the strong claim in Section 2 and the broader class of vanishing-Q examples in Section 3. Second, the phrase 'unit metric M' is imprecise, since M is then used algebraically as a scalar mass parameter. These are patchable presentation issues rather than core contradictions.

The main mathematical derivation in Section 2 is valid. Starting from ψ_j = √ρ_j e^{iφ_j/ℏ}, the gradient and Laplacian are expanded correctly by standard product and chain rules: ∇ψ_j = (∇√ρ_j + (i/ℏ)√ρ_j∇φ_j)e^{iφ_j/ℏ}, and ∇²ψ_j = [∇²√ρ_j + (2i/ℏ)∇√ρ_j·∇φ_j + (i/ℏ)√ρ_j∇²φ_j - (1/ℏ²)√ρ_j(∇φ_j)^2]e^{iφ_j/ℏ}. Substituting into the Schrödinger equation and separating real and imaginary parts yields, respectively, the continuity equation and the quantum Hamilton–Jacobi equation with the Bohm/Madelung quantum potential. This is a standard and correct calculation, and it directly supports the paper's central objection.

The score is not 5 because several consequential claims are asserted rather than fully proved. Most importantly, the paper says 'without loss of generality' it restricts to a scalar potential with unit metric/scalar M, but does not derive the corresponding result for the more general setting of the target paper. This does not invalidate the displayed derivation, but it leaves a gap in the claimed scope of the rebuttal. In addition, Section 3's analysis of the target paper's examples is compressed: the double-slit density model is schematic, and the claims that the harmonic-oscillator and hydrogen examples import quantum eigenfunctions through initial conditions are plausible but not mathematically established in detail from the cited constructions. These gaps affect the breadth and some application-specific criticisms, but they do not overturn the central mathematical point that omitting spatial amplitude derivatives removes the quantum potential and thus cannot justify an exact general equivalence.

For a comment paper, the central claim is quite testable: one can directly substitute the disputed ansatz into the Schrödinger equation and check whether the Laplacian acting on √ρ generates the quantum-potential term. This provides a clear criterion for refuting either the target paper's exact-equivalence claim or the present comment's objection. The paper also gives discriminating conditions: if ∇²√ρ/√ρ is nonzero for a proposed exact example, then the original classical-action-only derivation cannot be exact as stated; if examples succeed only when Q vanishes or when quantum eigenfunction content is encoded in the initial data, that supports the rebuttal. The limitation is that the submission is not itself proposing a new physical framework with multiple novel quantitative observational predictions; its testability is mainly mathematical/computational and comparative rather than experimental. Still, as a critique of an exactness claim, it is clearly falsifiable in principle and in practice.

The submission is very clear. It states the disputed claim, identifies the alleged mathematical mistake, performs the relevant substitution in a compact but readable way, and explains the consequence in standard language familiar to graduate-level readers. The organization is strong: introduction, mathematical correction, then diagnosis of why the original paper's examples may appear successful. Notation is consistent, and the distinction between total derivatives along trajectories and spatial derivatives in the Schrödinger PDE is communicated sharply. The only minor caveat is that the example-specific critiques are more programmatic than fully worked through in exhaustive technical detail, but that does not materially obstruct comprehension of the main argument.

The core mathematical point—that Madelung substitution yields a quantum-potential term unless amplitude derivatives vanish—is standard and not novel by itself. Likewise, identifying omission of that term as equivalent to a semiclassical/WKB-type truncation is established physics. The paper's contribution lies in applying this standard diagnostic specifically to the Lohmiller-Slotine construction and organizing their examples into two failure modes: special geometries with vanishing Q, and apparent recovery of quantum results via quantum basis information hidden in initial conditions. That targeted critical synthesis has value and is specific, but it is not a new mechanism or new framework. Thus the work is moderately original as a focused rebuttal, not highly original at the level of theoretical invention.

The argument is fully developed with all necessary mathematical steps shown. The paper: (1) clearly states the error in the original work (neglecting spatial derivatives), (2) provides the complete correct derivation including the quantum potential term, (3) systematically explains why each example in the original paper appears to work despite the error, (4) addresses both trivial density cases and the circular reasoning in bound state examples, and (5) places the critique in proper historical context with appropriate references to Madelung and Bohm. All variables are defined, boundary conditions are addressed through the specific examples, and the limitations of the original approach are explicitly stated.